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Math  /  Geometry

QuestionFor each equation, identify the vertex, focus, axis of symmetry, and directrix. Then graph the parabola.
1. (x−3)2=12(y−7)(x-3)^{2}=12(y-7)
2. (x+1)2=−12(y−6(x+1)^{2}=-12(y-6
3. (y−4)2=20(x+2)(y-4)^{2}=20(x+2)
4. −1(x+7)=(y+5)2-1(x+7)=(y+5)^{2}
5. (x+8)2=8(y−3)(x+8)^{2}=8(y-3)
6. −40(x+4)=(y−9)-40(x+4)=(y-9)
7. (y+5)2=24(x−1)(y+5)^{2}=24(x-1)
8. 2(y+12)=(x−6)22(y+12)=(x-6)^{2}
9. −4(y+2)=(x+8)2-4(y+2)=(x+8)^{2}
10. 10(x+11)=(y+3)210(x+11)=(y+3)^{2}

Studdy Solution

STEP 1

What is this asking? We need to find key features of a bunch of parabolas, like their vertex, focus, axis of symmetry, and directrix, and then sketch them! Watch out! Remember the signs in the parabola formulas!
A little slip-up can lead to a totally different graph.
Also, don't mix up the xx and yy values for the vertex, focus, and directrix.

STEP 2

1. Understand the Parabola Form
2. Find the Vertex
3. Calculate "p"
4. Determine Focus and Directrix
5. Identify Axis of Symmetry
6. Sketch the Parabola

STEP 3

Let's take a general look at the equation form we're dealing with.
We've got two main forms: (x−h)2=4p(y−k)(x-h)^2 = 4p(y-k) and (y−k)2=4p(x−h)(y-k)^2 = 4p(x-h).
The first one opens up or down, and the second one opens left or right.

STEP 4

The **vertex** is the tip of our parabola.
It's super easy to find!
In both forms, the vertex is at the point (h,k)(h, k).
Just remember to switch the signs of the numbers you see in the equation.

STEP 5

The value of **"p"** tells us how far the **focus** and **directrix** are from the **vertex**.
We find it by comparing our equation to the general form.
If we have (x−h)2=4p(y−k)(x-h)^2 = 4p(y-k), then the number multiplying (y−k)(y-k) is equal to 4p4p.
If we have (y−k)2=4p(x−h)(y-k)^2 = 4p(x-h), then the number multiplying (x−h)(x-h) is equal to 4p4p.
Solve for pp by dividing the number by **4**.

STEP 6

If the parabola opens up or down ((x−h)2=4p(y−k)(x-h)^2 = 4p(y-k)), the **focus** is at (h,k+p)(h, k+p), and the **directrix** is the horizontal line y=k−py = k-p.
If it opens left or right ((y−k)2=4p(x−h)(y-k)^2 = 4p(x-h)), the **focus** is at (h+p,k)(h+p, k), and the **directrix** is the vertical line x=h−px = h-p.

STEP 7

The **axis of symmetry** is a line that cuts the parabola perfectly in half.
If the parabola opens up or down, the axis of symmetry is the vertical line x=hx = h.
If it opens left or right, it's the horizontal line y=ky = k.

STEP 8

Now we have all the pieces!
Plot the **vertex**, draw the **axis of symmetry**, mark the **focus**, and sketch the **directrix**.
Then, draw a smooth curve through the vertex, making sure it hugs the focus and moves away from the directrix.

STEP 9

For each equation, I'm not going to write out every single tiny step, but you should when you're solving them!
Here are the key features:
1. Vertex: (3,7)(3, 7), Focus: (3,10)(3, 10), Directrix: y=4y = 4, Axis: x=3x = 3
2. Vertex: (−1,6)(-1, 6), Focus: (−1,3)(-1, 3), Directrix: y=9y = 9, Axis: x=−1x = -1
3. Vertex: (−2,4)(-2, 4), Focus: (3,4)(3, 4), Directrix: x=−7x = -7, Axis: y=4y = 4
4. Vertex: (−7,−5)(-7, -5), Focus: (−7.25,−5)(-7.25, -5), Directrix: x=−6.75x = -6.75, Axis: y=−5y = -5
5. Vertex: (−8,3)(-8, 3), Focus: (−8,5)(-8, 5), Directrix: y=1y = 1, Axis: x=−8x = -8
6. Vertex: (−4,9)(-4, 9), Focus: (−14,9)(-14, 9), Directrix: x=6x = 6, Axis: y=9y = 9
7. Vertex: (1,−5)(1, -5), Focus: (7,−5)(7, -5), Directrix: x=−5x = -5, Axis: y=−5y = -5
8. Vertex: (6,−12)(6, -12), Focus: (6,−11.5)(6, -11.5), Directrix: y=−12.5y = -12.5, Axis: x=6x = 6
9. Vertex: (−8,−2)(-8, -2), Focus: (−8,−3)(-8, -3), Directrix: y=−1y = -1, Axis: x=−8x = -8
10. Vertex: (−11,−3)(-11, -3), Focus: (−8.5,−3)(-8.5, -3), Directrix: x=−13.5x = -13.5, Axis: y=−3y = -3

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